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Current Question (ID: 9571)

Question:

\text{The following are four different relations about displacement, velocity, and acceleration for the motion of a particle in general. Choose the incorrect statement(s):} \text{a. } v_{\text{avg}} = \frac{1}{2}[v(t_1) + v(t_2)] \text{b. } v_{\text{avg}} = \frac{r(t_2) - r(t_1)}{t_2 - t_1} \text{c. } r = \frac{1}{2}[v(t_2) - v(t_1)](t_2 - t_1) \text{d. } a_{\text{avg}} = \frac{v(t_2) - v(t_1)}{t_2 - t_1}

Options:

1. $\text{(a), (d)}$
2. $\text{(a), (c)}$
3. $\text{(b), (c)}$
4. $\text{(a), (b)}$

Solution:

$\text{Hint: Recall the formulas of average velocity and acceleration.}$ $\text{Step 1: Check all options one by one}$ $v_{avg} = \frac{v(t_1) + v(t_2)}{2}$ $\text{This is only valid for uniform acceleration, not general motion}$ $v_{avg} = \frac{\Delta \vec{r}}{\Delta t} = \frac{r(t_2) - r(t_1)}{t_2 - t_1}$ $\text{This is the correct definition for any motion}$ $a_{avg} = \frac{v(t_2) - v(t_1)}{t_2 - t_1}$ $\text{This is the correct definition of average acceleration}$ $\text{Statement (c) does not represent any valid kinematic relation for general motion}$ $\text{Therefore, statements (a) and (c) are incorrect for general motion.}$

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Expected JSON Format:

{
  "question": "The mass of carbon present in 0.5 mole of $\\mathrm{K}_4[\\mathrm{Fe(CN)}_6]$ is:",
  "options": [
    {
      "id": 1,
      "text": "1.8 g"
    },
    {
      "id": 2,
      "text": "18 g"
    },
    {
      "id": 3,
      "text": "3.6 g"
    },
    {
      "id": 4,
      "text": "36 g"
    }
  ],
  "solution": "\\begin{align}\n&\\text{Hint: Mole concept}\\\\\n&1 \\text{ mole of } \\mathrm{K}_4[\\mathrm{Fe(CN)}_6] = 6 \\text{ moles of carbon atom}\\\\\n&0.5 \\text{ mole of } \\mathrm{K}_4[\\mathrm{Fe(CN)}_6] = 6 \\times 0.5 \\text{ mol} = 3 \\text{ mol}\\\\\n&1 \\text{ mol of carbon} = 12 \\text{ g}\\\\\n&3 \\text{ mol carbon} = 12 \\times 3 = 36 \\text{ g}\\\\\n&\\text{Hence, 36 g mass of carbon present in 0.5 mole of } \\mathrm{K}_4[\\mathrm{Fe(CN)}_6].\n\\end{align}",
  "correct_answer": 4
}